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50
Extended gaussian images
 Proceedings of the IEEE
, 1984
"... This is a primer on extended Gaussian Images. Extended Gaussian Images are useful for representing the shapes of surfaces. They can be computed easily from: 1. Needle maps obtained using photometric stereo, or 2. Depth maps generated by ranging devices or stereo. Importantly, they can also be determ ..."
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Cited by 210 (3 self)
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This is a primer on extended Gaussian Images. Extended Gaussian Images are useful for representing the shapes of surfaces. They can be computed easily from: 1. Needle maps obtained using photometric stereo, or 2. Depth maps generated by ranging devices or stereo. Importantly, they can also be determined simply from geometric models of the objects. Extended Gaussian images can be of use in at least two of the tasks facing a machine vision system.
(MIT AI Memo 740)
Representation and Recognition of FreeForm Surfaces
, 1992
"... We introduce a new surface representation for recognizing curved objects. Our approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ..."
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Cited by 62 (7 self)
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We introduce a new surface representation for recognizing curved objects. Our approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. We define local regularity constraints that the mesh must satisfy. We then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is poseinvariant is stored at every node of the mesh. We use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. We show how the similarity between reference model and observed data can be evaluated and we show how the pose of the reference object in the observed scene can be easily computed using this representation. We present results on real range images which show that this approach to modelling and recognizing threedimensional objects has three main advantages: First, it is applicable to complex curved surfaces that cannot be handled by conventional techniques. Second, it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search. Finally, even though it is based on a spherical mapping, the approach can handle occlusions and partial views.
Recognition of 3d objects using the extended gaussian image
 In IJCAI Conference
, 1981
"... propose to use an extended Gaussian imap,e (EGI) for interpreting 21/2D representations for recognition of 3D objects. The EGI is constructed by mapping each surface normals of an object to the Gaussian sphere. The freedom in viewer directions caused by incomplete observation Is greatly reduced b ..."
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Cited by 36 (14 self)
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propose to use an extended Gaussian imap,e (EGI) for interpreting 21/2D representations for recognition of 3D objects. The EGI is constructed by mapping each surface normals of an object to the Gaussian sphere. The freedom in viewer directions caused by incomplete observation Is greatly reduced by applying constraints derived from a global distribution of surface normals on the EGI. One constraint on the viewer direction is derived from the ratio of the projected area to the original surface area. The other constraint comes from the direction of the principal axis. After reducing the possible viewing directions with these constraints, we will apply a matching function to ESls of a candidate set for a final decision. We also propose an algorithm for reconstruction of the original shape of a convex polyhedron from its EGI. This algorithm is based on the analysisbysynthesis method. 1 WHAT IS THE EXTENDED GAUSSIAN IMAGE A collection of local surface normals [1,2,3,4,5], sometimes referred to as a 21/2D representation of an object [6], is often provided by machine vision at the low level. For example, an algorithm based on the propagationofconstraints technique [2] provides local surface orientation from shading and occluding information. The same algorithm can also produce surface orientation from apparent distortion of known patterns based on a regularpattern gradient map [A], The distortion of these small circles on the golf ball in Fig. 1 can be used to recover local surface orientation.
Ununfoldable polyhedra with convex faces
 COMPUT. GEOM. THEORY APPL
, 2002
"... Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex fa ..."
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Cited by 26 (10 self)
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Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.
Determining a depth map using a dual photometric stereo
 The International Journal of Robotics Research
, 1987
"... This paper describes a method for determining a depth map from a pair of surfaceorientation maps obtained by a dual photometric stereo. A photometric stereo system determines surface orientations by taking three images from the same position under different lighting conditions, based on the shading ..."
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Cited by 20 (4 self)
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This paper describes a method for determining a depth map from a pair of surfaceorientation maps obtained by a dual photometric stereo. A photometric stereo system determines surface orientations by taking three images from the same position under different lighting conditions, based on the shading information. A photometric stereo system can determine surface orientations very rapidly, but cannot determine absolute depth values. This paper proposes a dual photometric stereo system to obtain absolute depth values. A dual photometric stereo generates a pair of surfaceorientation maps. Then, the surfaceorientation maps can be segmented into isolated regions with respect to surface orientations, using a geodesic dome for grouping surface orientations. The resulting left and right regions are compared to pair corresponding regions. The following three kinds of constraints will be used to search for corresponding regions efficiently: a surfaceorientation constraint, an area constraint, and an epipolar constraint. Region matching is done iteratively, starting from a coarse segmented result and proceeding to a fine segmented result, using a parentchildren constraint. The horizontal difference in the position of the center of mass of a region pair determines the absolute depth value for the physical surface patch imaged onto that pair. This system takes only a few minutes on a Lisp machine to determine an absolute depth map in complicated scenes and could be used as an input device for a binpicking system. 1.
Creating Polyhedral Models by Computer
 Journal of Computers in Mathematics and Science Teaching
, 1997
"... This paper describes a computer application named HyperGami that permits users to design, explore, decorate, and study a rich variety of paper polyhedral models. In structure, HyperGami is a "programmable design environment", including both a direct manipulation interface as well as a doma ..."
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Cited by 16 (7 self)
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This paper describes a computer application named HyperGami that permits users to design, explore, decorate, and study a rich variety of paper polyhedral models. In structure, HyperGami is a "programmable design environment", including both a direct manipulation interface as well as a domainenriched programming environment based on the Scheme language; the application is thus designed to be accessible to students of geometry while providing challenging projects for longterm or expert users (such as professional mathematicians and designers). In the course of this paper, we describe the HyperGami interface and language; illustrate the construction of "customized polyhedra" of various sorts; discuss the results of our initial experiences using the system in working with middleschool students; and argue for the utility of embedding programming languages in educational design environments such as this one. 1. Introduction Over the centuries, human beings have been fascinated by polyhe...
Classification of tilings of the 2dimensional sphere by congruent triangles
 Hiroshima Math. J
, 2002
"... We give a new classification of tilings of the 2dimensional sphere by congruent triangles accompanied with a complete proof. This accomplishes the old classification by Davies, who only gave an outline of the proof, regrettably with some redundant tilings. We clarify Davies ’ obscure points, give a ..."
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Cited by 15 (0 self)
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We give a new classification of tilings of the 2dimensional sphere by congruent triangles accompanied with a complete proof. This accomplishes the old classification by Davies, who only gave an outline of the proof, regrettably with some redundant tilings. We clarify Davies ’ obscure points, give a complete list, and show that there exist ten sporadic and also ten series of such tilings, including some unfamiliar twisted ones. We also give their figures, development maps in a way easy to understand their mutual relations. In Appendix, we give curious examples of tilings on noncompact spaces of constant positive curvature with boundary possessing a special 5valent vertex that never appear in the tiling of the usual sphere.
Continuous Indexing of Hierarchical Subdivisions of the Globe
 International Journal of Geographical Information Science
, 2000
"... We describe how to create a continuous global index of the surface of the earth. The model is based on a hierarchical subdivision of the surface into triangular regions, in which eachregion is assigned a numerical label according to a spacefilling curve. Sequential labels are assigned to adjacent ..."
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Cited by 14 (3 self)
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We describe how to create a continuous global index of the surface of the earth. The model is based on a hierarchical subdivision of the surface into triangular regions, in which eachregion is assigned a numerical label according to a spacefilling curve. Sequential labels are assigned to adjacent regions, so labels can be sorted to create a continuous onedimensional index. Benefits of this continuous model include the implicit preservation of adjacency information, and the abilityto vary resolution at different locations. Previously suggested schemes based on similar models produce indices that are discontinuous.
Ununfoldable Polyhedra
, 1999
"... A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can inde ..."
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Cited by 14 (9 self)
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A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.